Extreme-value analysis in nano-biological systems: applications and implications

Abstract

Extreme value analysis (EVA) is a statistical method that studies the properties of extreme values of datasets, crucial for fields like engineering, meteorology, finance, insurance, and environmental science. EVA models extreme events using distributions such as Fréchet, Weibull, or Gumbel, aiding in risk prediction and management. This review explores EVA’s application to nanoscale biological systems. Traditionally, biological research focuses on average values from repeated experiments. However, EVA offers insights into molecular mechanisms by examining extreme data points. We introduce EVA’s concepts with simulations and review its use in studying motor protein movements within cells, highlighting the importance of in vivo analysis due to the complex intracellular environment. We suggest EVA as a tool for extracting motor proteins’ physical properties in vivo and discuss its potential in other biological systems. While there have been only a few applications of EVA to biological systems, it holds promise for uncovering hidden properties in extreme data, promoting its broader application in life sciences.

Full text

Vol.:(0123456789)
Biophysical Reviews
https://doi.org/10.1007/s12551-024-01239-w
REVIEW
Extreme‑value analysis innano‑biological systems: applications
andimplications
KumikoHayashi1· NobumichiTakamatsu1· ShunkiTakaramoto1
Received: 31 July 2024 / Accepted: 24 September 2024
© The Author(s) 2024
Abstract
Extreme value analysis (EVA) is a statistical method that studies the properties of extreme values of datasets, crucial for
fields like engineering, meteorology, finance, insurance, and environmental science. EVA models extreme events using
distributions such as Fréchet, Weibull, or Gumbel, aiding in risk prediction and management. This review explores EVA’s
application to nanoscale biological systems. Traditionally, biological research focuses on average values from repeated
experiments. However, EVA offers insights into molecular mechanisms by examining extreme data points. We introduce
EVA’s concepts with simulations and review its use in studying motor protein movements within cells, highlighting the
importance of invivo analysis due to the complex intracellular environment. We suggest EVA as a tool for extracting motor
proteins’ physical properties invivo and discuss its potential in other biological systems. While there have been only a few
applications of EVA to biological systems, it holds promise for uncovering hidden properties in extreme data, promoting
its broader application in life sciences.
Keywords Motor proteins· Kinesin· Dynein· Extreme value statistics
Introduction
Extreme value analysis (EVA) is a branch of statistics that
focuses on extreme values. It is the study of the statisti-
cal properties of particularly large or small values within
a data set. EVA is widely applied in various fields where
extreme phenomena hold significant importance (Coles
2001; Gilleland and Katz 2016), such as disaster prevention
(de Haan and Ferreira 2006; Tippett etal. 2016), finance
(Kratz 2019), safety estimation (Songchitruksa and Tarko
2006), sports (Einmahl and Magnus 2008; Gembris etal.
2002), human lifespan (Dong etal. 2016; Rootzen and Zolud
2017), and the recent pandemic (Wong and Collins 2020).
Recently, its applications in the biological data analysis has
also become active (Basnayake etal. 2019; Tsuduki 2024).
The central concepts of EVA involve identifying the largest
or smallest values in a dataset, determining whether these
extremes follow distributions like the Fréchet distribution,
Weibull distribution, or Gumbel distribution, and analyzing
data points that exceed specific thresholds to estimate their
distribution. EVA is utilized in a broad range of fields. For
instance, in meteorology, it is used to predict and assess
extreme weather events such as typhoons, floods, and
droughts. In finance, it aids in risk management for extreme
price fluctuations, such as market crashes or surges. In engi-
neering, it is applied to analyze extreme stresses or loads
for evaluating the durability and safety design of structures.
EVA serves as a powerful tool for predicting the frequency
and impact of extreme events, aiding in risk management
and safety measures.
In this review paper, we discuss the application of EVA
to the analysis of experimental data in nanoscale bio-
logical systems. By using EVA, it is possible to explore
novel molecular mechanisms in life sciences through the
behavior of extreme values in the data. Rare events in
nanoscale biological systems, such as protein misfolding
(Harada etal. 2015) and synaptic delay (Tsuduki 2024),
have already been studied, and their prediction is consid-
ered important due to their relevance to diseases. In the
following sections, we first learn the concepts of EVA
through simple simulations using random numbers. Next,
we introduce the application of EVA to the movement
* Kumiko Hayashi
hayashi@issp.u-tokyo.ac.jp
1 The Institute forSolid State Physics, The University
ofTokyo, Kashiwano-Ha 5-1-5, Kashiwa, Chiba277-8581,
Japan

Biophysical Reviews

Biophysical Reviews
of motor proteins within cells, investigated in our recent
research paper (Naoi etal. 2024). The physical properties
of motor proteins, such as force and velocity, have been
investigated by invitro single-molecule experiments, in
which the functions of motor proteins consisting of mini-
mal complexes were analyzed in glass chambers (Bren-
ner etal. 2020; Elshenawy etal. 2019; Gennerich etal.
2007; Hirakawa etal. 2000; Mallik etal. 2004; Schnitzer
etal. 2000; Toba etal. 2006). However, because motor
proteins function fully in the intracellular environment
and are equipped with accessory proteins, the investiga-
tion of motor proteins invivo is as important as invitro
single-molecule experiments. We believe that EVA can
be useful as a new information science tool for extracting
the physical properties of motor proteins in the complex
intracellular environment.
Finally, we discuss the potential applications of EVA in
nanoscale biological systems, with a specific example of its
application to the analysis of droplet sizes in liquid–liquid
phase separation (LLPS) (Takaramoto and Inoue 2024). This
application of EVA was inspired by Takaramoto and Inoue’s
poster at the IUPAB 2024 Congress. While research utiliz-
ing EVA in nanoscale biological systems is still limited, it is
believed that undiscovered properties may lie hidden within
the extreme values of experimental data.
Simulation using random numbers
In this section, the analytical methods of EVA are reviewed
through a simple simulation using uniform random num-
bers ranging from 0 to 1 and Gaussian random numbers.
In the case of random numbers ranging from 0 to 1, it is
obvious that the finite maximum value is 1, while Gaussian
random numbers do not have a finite maximum value. In
other words, we examine two random numbers with different
extreme properties.
One block of EVA is considered as a data set (i.e., a block)
of elements
M
, from which the largest value is selected.
In meteorology,
M
is often considered as 1year, which is
365days. The selection of the block maximum (
Xi
max
) from
the
i
-th block (the
i
-th data set) is repeated
n
times to col-
lect the block maximum data set
{
Xi
max}
(
i=1,
⋯
,n
). Using
the block maximum data set
{
X
i
max}
, the return level plot is
investigated (the plot was calculated by using the ismev and
evd packages in R (R Core Team 2018)) (Fig.1a and b).
The two axes of the return level plot (Fig.1a and b, bottom)
represent the return period
rp
and return level
zp
. For a given
probability
p
,
rp
=−
{log(1−p)}
−1
, and
zp
is defined by the
generalized extreme value distribution as
1−p=G(zp)
,
where
Note that
zp
represents

X
i
max
, where
{

Xi
max
}
is the rear-
ranged data of
{
X
i
max}
, such that

X
1
max
≤

X2
max
≤⋯≤

Xn
max
.
Roughly,
rp
is the sample number. (Note that we obtained
parameters of the generalized extreme value distribution
𝜉
,
𝜇
, and
𝜎
in Eq.(1) by using the ismev and evd packages in
R(R Core Team 2018).) Here, the return level
zp
refers to the
magnitude of extreme values expected to occur once within
the sample size we are considering, such as 100years, 1000
people, and so on. For example, when analyzing river water
level data for environmental science case, its return level plot
can provide information such as the maximum water level
of a flood expected once every 100years. The distribution
characteristics of the extreme value data, such as the values
of location (
𝜇
), scale (
𝜎
), and shape parameters (
𝜉
) of the
extreme value distribution (the Fr e chet distribution,
Weibull distribution, or Gumbel distribution) can be esti-
mated from the behavior of the return level plot (Fig.1c).
The shape parameters of the extreme value distribution
𝜉
are important because it characterizes the behavior of
extreme values, determining whether a data set has a finite
maximum value or not. We found that
𝜉<0
for the case
of uniform random numbers, which have the finite maxi-
mum value (Fig.1a), and
𝜉∼0
for the case of Gaussian
random numbers, which do not have the finite maximum
value (Fig.1b). The existence of a finite maximum value is
related to the value of the shape parameter
𝜉
. Therefore, we
particularly focus on the value of
𝜉
in EVA.
Depending on the value of
𝜉
, it is classified into the
Weibull distribution (
𝜉<0
), Gumbel distribution (
𝜉=0
),
or Fr e chet distribution (
𝜉>0
) (Fig.1c). In the case of a
Weibull distribution, the finite extreme value (the maximum
value)
Vex
is proved to exist and is estimated using the fol-
lowing equation:
It has been known that the extreme values (the maxi-
mum values) of a uniform random number distribution
converge to a Weibull distribution (Fig.1a), and those of
a Gaussian distribution to Gumbel distribution (Fig.1b).
In our simulations, we obtained the set of parameters
(𝜇,𝜎,𝜉)=(0.911 ±0.003SE, 0.0793 ±0.0034SE, −0.890 ±0.012SE)
for uniform random numbers ranging from 0 to 1. For the
(1)
G
zp

=exp

−

1+𝜉

zp−𝜇
𝜎

−1∕
𝜉.
(2)
Vex =𝜇−𝜎∕𝜉
Fig. 1 Simulation using random numbers. The cases for uniform ran-
dom numbers ranging from 0 to 1 (
M=10
,
n=500
) (a) and Gauss-
ian random numbers (
M=1000
,
n=500
) (b). Each panel represents
the counts, cumulative probability, and return level plot for the block
maximum. The shape parameter
𝜉<0
for the uniform random num-
bers and
𝜉∼0
for the Gaussian random numbers. Red lines represent
G(Xmax)
(Eq.(1)). Examples of extreme value distributions (c)
◂

Biophysical Reviews
set of parameters
(𝜇,𝜎,𝜉)
, the quantity
𝜇−𝜎∕𝜉
(Eq.(2)) was
calculated to be 1.00 as the maximum number.
The properties of the maximum value of the simulation
examples shown in Fig.1 are known from the beginning.
However, the important point is that the extreme value distri-
bution of any data, whose extreme properties are not known,
can be described by Eq.(1), based on the mathematical theo-
rem. In unknown complex systems like nanoscale biological
systems, it is a significant advantage to know the functional
form (Eq.(1)) for their extreme value distributions. In the
next section, we consider the application of this strength in
EVA to intracellular motor proteins.
Applications ofEVA tocargo transport
bymotor proteins
Motor protein is a general term for proteins that move and
function using energy obtained from adenosine triphosphate
(ATP) hydrolysis. Here, we focus on kinesin and dynein
among such motor proteins, which are responsible for
cargo transport and form the basis of intracellular material
logistics (Fig.2a). Particularly, we investigate anterograde
transport by kinesin and retrograde transport by dynein
in the axons of neurons. In the logistics of neurons, both
anterograde transport, which carries synaptic materials to
the terminals, and retrograde transport, which collects waste
materials, are important. Because logistics in long axons
is particularly important for neurons, and disruptions in
this process are often associated with neurological diseases
(Guedes-Dias,Holzbaur 2019; Keefe etal. 2023), the appli-
cation of EVA to axonal transport is considered to be sig-
nificant to help elucidate disease mechanisms in the future
by increasing invivo physical quantities we can estimate.
Cargo motion can be observed using fluorescence micros-
copy (Fig.2a) (Hayashi etal. 2018; Naoi etal. 2024). Then,
the transport velocity is measured by analyzing the recorded
movies obtained from the fluorescence microscopy. Note
that fluorescence observation is a standard technique in bio-
physics, and this review will not discuss the methods of fluo-
rescence observation. In invivo cases, the measured velocity
values varied affected by the size of a cargo and the number
of motor proteins involved in the cargo transport (Fig.3a).
Especially since cargo sizes can differ by a factor of 10, the
average velocity value reflects the cargo size rather than the
performance of the motor proteins due to high viscosity in
cells, unlike the situations in invitro single-molecule experi-
ments. The differences in the mechanical properties of the
two different motor proteins, kinesin and dynein, are also
less likely to be reflected in the velocity values, because the
influence of cargo size on the velocity values is larger than
the differences in the two motors. Then, the difference and
true performance of the motor proteins can be considered to
appear in the limit of small cargo sizes. Under this condi-
tion, it is believed that the difference and true performance
of motor proteins can be accurately evaluated using EVA. In
this review, in the following, we introduce the application of
EVA to axonal transport in two different species: C. elegans
worms and mice.
C. elegans worms
The measurement method of transport velocity using
fluorescence imaging in living C. elegans worms was
explained in the original paper (Naoi etal. 2024), not-
ing that fluorescent proteins were labeled on the cargo
for the fluorescence microscopy (Fig.2a). Approximately
2000 velocity data were collected from about 200 worms
for anterograde and retrograde transport. One thing to be
careful about is the fact that a measurement difficulty in
live worms caused the small size of
M
(
M=10
for this
experiment), which is typically set to of the order of 1000
to obtain a correct extreme value distribution, highlight-
ing a key problem in the application of EVA to nanoscale
biological systems, which is discussed in the “Discussion”
section.
Using the block maximum dataset
{
vi
max}
(
i=1,
⋯
,n
where
n
=228 for anterograde transport and
n
=217 for
retrograde transport), the return-level plot was calculated
(it was calculated concretely by using the ismev and evd
packages in R (R Core Team 2018)) (Fig.2b).
𝜉<0
for
anterograde transport and
𝜉∼0
for retrograde transport.
For anterograde transport, the return level plot shows a
convergent behavior as
rp
becomes larger, a property spe-
cific to a Weibull distribution (
𝜉<0
), and the extreme
value
Vex
was estimated to be 4.0 ± 0.4μm/s using Eq.(2).
Because the return-level plot of the retrograde transport
(Fig.2b) shows
𝜉∼0
,
Vex
cannot be estimated from Eq.(2)
for retrograde transport by dynein, noting that the finite
maximum exists only for the case
𝜉<0
. In the follow-
ing simulation section, we explore why dynein does not
exhibit a behavior of Weibull distribution.
Mice
To investigate a mammalian case, EVA was also applied
to examine the velocities of synaptic cargo transport by
motor proteins in mouse hippocampal neurons (Fig.2c,
left), as originally reported in our previous study (Hayashi
etal 2021). The return-level plot with
𝜉<0
was observed
for anterograde transport. Then, the maximum velocity
for anterograde transport in mice hippocampal neurons
was calculated to be 5.6 ± 1.4 μm/s using Eq.(2). The
maximum velocity was higher than that of the worms.

Biophysical Reviews
Fig. 2 Schematics of synap-
tic cargo transport by motor
proteins in the axon of a
neuron. a A cargo is antero-
gradely transported by KIF1A
(kinesin) and retrogradely by
cytoplasmic dynein. b, c Return
level plot and histogram of the
block maximum velocity data
of synaptic cargo transport
by motor proteins. Results for
anterograde (red) and retrograde
(blue) transport, in the cases of
the motor neurons of C. elegans
worms (Naoi etal. 2024) (b),
and the hippocampal neurons of
mice (Hayashi etal. 2021) (c)

Biophysical Reviews

Biophysical Reviews
Comparing the maximum velocities among various spe-
cies is a future important issue. Return-level plot with
𝜉>0
was also observed for retrograde transport (Fig.2c,
right). It is known that dynein exhibits different mechani-
cal properties depending on the species, but it is expected
to have similar mechanical properties in both worms and
mice, judging from the similar behaviors of the return
level plots.
Numerical simulation
In our previous paper on the application of EVA to motor pro-
teins (Naoi etal. 2024), the opposite signs of
𝜉
shown in the
return level plots (Fig.2b, c) was attributed to the different
force (load)-velocity relationship between kinesin and dynein.
The difference in the convexity of force (load)-velocity rela-
tionships (Fig.3b), which have been clarified based on invitro
single-molecule studies using optical tweezers, reflects the dif-
ferent mechanisms underlying the walking behavior of motor
proteins on microtubules, although the biological meanings of
convexity are not clearly understood yet. Due to high viscos-
ity in cells, a motor protein experiences a load proportional
to the cargo size transported by it. Therefore, we predicted
that the behavior of velocity values invivo would be influ-
enced by the force (load)-velocity relationship. The observed
extreme velocity values, which are close to those under the
zero-load condition, are particularly related to the properties
of the force (load)-velocity relationship under low load condi-
tion areas (green and yellow area depicted in Fig.3b, right).
The property that
𝜉∼0
for the retrograde velocity data was
attributed to the fact that the force (load)-velocity relation-
ship was concave. The steep velocity decrease for the case of
the concave force (load)-velocity relationship in the low-load
condition (green area in Fig.3b) caused a major variation in
the larger velocity values, and this behavior tends to generate a
major variation in velocity. Then large gaps generated between
vi
sim, max
and
vi+1
sim, max
for a large
i
in this case. This gap caused
the non-convergence of
vi
max
and
𝜉∼0
as a result. In the first
place, the occurrence of small cargos, which are thought to
generate high velocities, is a rare event. As a result, in a con-
cave force (load)-velocity relationship, large velocity values
are less likely to converge due to this rarity.
We can actually reproduce the behavior observed for anter-
ograde and retrograde transport using a theoretical model of
the force (load)-velocity relationship derived based on the
mechanisms underlying ATP hydrolysis by motor proteins
(Sasaki etal. 2018). The force (load)-velocity relationship for
the model is represented as follows:
See reference (Sasaki etal. 2018) for the definitions and
values of parameters. The differences in the model param-
eters (Eq.(3)) for kinesin and dynein were resulted in the
different convexities of the force–velocity relationship. Fig-
ure3c (top) represents the
{
vi
sim}
(4000 data) obtained from
the simulations using the model (Eq.(3)).
{
vi
sim, max
}
were
chosen from
{
v
i
sim}
(
M=10
). Using these
{
vi
sim, max
}
(
n=400
), we calculated the return-level plots for kinesin
and dynein (Fig.3c, bottom). We found that the tendency
that
zp
(
=
v
i
sim, max
arranged in ascending order) did not con-
verge for a large
rp
(roughly,
rp
is the sample number (1
≤rp≤
400)) in the case of dynein, i.e., the gaps created
between
vi
sim, max
and
vi+1
sim, max
for a large
i
. This is because a
large velocity value is likely to be generated in the case of a
concave force (load)-velocity relationship, owing to its steep
slope in the load-sensitive regime.
Discussion
Typically, a block size (the number of elements in a block,
M
) of around 1000 is used to obtain a correct extreme value
distribution (Coles 2001). In comparison, our study uses an
extremely small
M
, around 10. This is due to the limited
cargo transport observable in a single C. elegans worm,
resulting in an
M
value of around 10. It is difficult to collect
a large number of samples within living organisms or cells.
This is a fundamental issue in biological systems. In our
research, we checked that while the parameters
𝜇
,
𝜎
, and
𝜉
showed dependency on
M
due to its small size, the qualita-
tive results of
𝜉<
0 for anterograde and
𝜉>
0 for retrograde
transport were independent of
M
(Naoi etal. 2024). Due to
the small sample size, it may be helpful to use other statisti-
cal methods, such as a bootstrap method, to cope with outli-
ers of data when we apply EVA (Naoi etal. 2024).
There is one thing that should be mentioned regarding
the conclusion obtained from the numerical simulation
depicted in Fig.3c. We attributed the property
𝜉∼0
in the
extreme value data for retrograde transport (specifically the
rare occurrence of large velocity values and the divergence
observed in the return level plot at large
rp
values) to a con-
cave force (load)-velocity relationship. However, several
mechanisms could lead to the occasional large velocity val-
ues in retrograde transport, such as active non-equilibrium
(3)
v
three(F)=
(
k01 −k02
)l
1
k01
=1
𝜅1
+1
𝜆1
ed1F∕kBT
1
k02
=1
𝜆2
e−d2F∕kBT
Fig. 3 Physical parameters of intracellular cargo transport by motor
proteins. a The variety in the values of cargo transport resulted from
the number of motor proteins and cargo sizes. b Load dependence of
velocity in convex and concave cases. An existence of a cargo can be
a load in the axons of neurons, because of high viscosity in the cells.
c Simulation results for the model (Eq.(3))
◂

Biophysical Reviews
fluctuations originating from the cellular cytoskeleton
(Chaubet etal. 2020; Mizuno etal. 2007) and the participa-
tion of multiple molecular motors (Fig.3a) (Rai etal. 2013).
In the future, we wish to further investigate the origin of
large velocity values observed for retrograde transport.
To expand the application of EVA in nanoscale bio-
logical systems, it is crucial to consider applying EVA to
systems entirely different from motor proteins. Inspired
by the poster presentation by Takaramoto and Inoue at the
IUPAB2024 Congress (Takaramoto and Inoue 2024), we
became interested in the analysis of droplet sizes formed
by liquid–liquid phase separation (LLPS). Within cells,
there are membrane-less organelles formed through LLPS,
such as stress granules and nucleoli. These organelles are
formed through local concentration and separation caused
by interactions between specific proteins and RNA, play-
ing critical roles in cellular functions. While membrane-
less organelles formed by LLPS are reversibly assembled
and disassembled under normal conditions, in pathological
states, this process can break down, leading to the for-
mation of irreversible aggregates. In neurodegenerative
diseases such as ALS and Parkinson’s disease, abnormal
irreversible aggregation of proteins has been reported to
cause neuronal cell death. Takaramoto and Inoue investi-
gated the mechanism of liquid condensate formation by the
Parkinson’s disease-related protein α-synuclein (αSyn).
A previous study (Hoffmann etal. 2021) has shown that
when αSyn coexists with the neuronal protein synapsin,
αSyn is incorporated into the droplets formed by synapsin
and can undergo phase separation at lower concentrations
than when αSyn forms droplets alone. Takaramoto and
Inoue studied the detailed molecular mechanism of droplet
formation by αSyn in the presence of synapsin. Takara-
moto and Inoue confirmed droplet formation in solution
of αSyn and synapsin (in vitro experiment) (Fig.1 of ref-
erence (Takaramoto and Inoue 2024)) and obtained the
droplet size distribution. From our preliminary applica-
tion of EVA to the droplet size distribution, we found that
this distribution follows the Fréchet type, which does not
have a finite maximum value (unpublished data). On the
other hand, within neurons, a maximum value for droplet
size is expected to exist due to the fixed cell size. It is
important to compare both the droplet formation in the
invitro experiment and cellular environments. In addi-
tion to experiments, molecular simulations have emerged
as powerful tools to elucidate the physicochemical prin-
ciples of biomolecular condensates, complementing
experimental approaches. At Joseph’s keynote session at
IUPAB2024, we learned the potential of LLPS-specific
simulations (Mpipi) (Joseph etal. 2021). Pursuing whether
the maximum value of LLPS droplet size is related to the
mechanism of disease onset and describing life sciences
through extreme values rather than averages is a crucial
task for the future.
Author contribution K.H. wrote the main manuscript text. N.T. and
S.T. contributed the part of discussion section.
Funding Open Access funding provided by The University of Tokyo.
Data availability Data sharing is not applicable to this article as no
new data were created.
Declarations
Conflict of interest The authors declare no competing interests.
Open Access This article is licensed under a Creative Commons Attri-
bution 4.0 International License, which permits use, sharing, adapta-
tion, distribution and reproduction in any medium or format, as long
as you give appropriate credit to the original author(s) and the source,
provide a link to the Creative Commons licence, and indicate if changes
were made. The images or other third party material in this article are
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permitted by statutory regulation or exceeds the permitted use, you will
need to obtain permission directly from the copyright holder. To view a
copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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